I think the reason might be because a tech can't diagnose anything, only a MD, or a midlevel practitioner. Reading your ultrasound and answering any questions that may come up can be considered giving a diagnosis. If the tech gives a wrong answer then all hell breaks loose. The hospital may be choosing to enforce this policy so the tech can't show you anything. Why can't you see the screen? Maybe to avoid potential questions "What's that?" "Is that my baby?" ect.
For example: I had a 17 year old who was hit by a car and broke his leg. He kept complaining of shoulder pain so two days into his admit an x-ray was done. I looked it up on the computer at work and it was a clean, very obvious break. Neither the radiologist or the attending had read the x-ray to diagnose him with a broken clavicle. I could not tell the mom anything without risking my job and my license, I can't give a medical diagnosis because I'd be "practicing medicine without a license" I practice nursing, not medicine. The only thing I could tell her was that the x-ray was up and waiting for interpretation.
I'm allowed to see my ultrasound at the hospital because the man was with me when I went. He's a mid-level in the ED so the techs know him, he has privileges in the hospital, he can diagnose and he's well trained to read an US.
ETA: I'm not defending the tech or the hospital policy I'm just giving a possible explanation.